This issue is covered by law (Tarasoff) and in this case given what we know so far, it does not rise to the level of "Duty to Warn" on the part of the doctor. If the police had wanted to they could have committed Holmes for psych evaluation on their own authority. But they didn't. "Doctor-patient privilege" is the same as lawyer-client privilege and the requirements for violating that privilege are specific and strict. They didn't happen here.